Standard Form of a Linear Programming Problem


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1 494 CHAPTER 9 LINEAR PROGRAMMING 9. THE SIMPLEX METHOD: MAXIMIZATION For linear programming problems involving two variables, the graphical solution method introduced in Section 9. is convenient. However, for problems involving more than two variables or problems involving a large number of constraints, it is better to use solution methods that are adaptable to computers. One such method is called the simplex method, developed by George Dantzig in 946. It provides us with a systematic way of examining the vertices of the feasible region to determine the optimal value of the objective function. We introduce this method with an example. Suppose we want to find the maximum value of z 4x 6x, where x and x, subject to the following constraints. x 5x x 5x x 5x 9 Since the lefthand side of each inequality is less than or equal to the righthand side, there must exist nonnegative numbers s, s and s that can be added to the left side of each equation to produce the following system of linear equations. x 5x s s s x 5x s s s x 5x s s s 9 The numbers s, s and s are called slack variables because they take up the slack in each inequality. Standard Form of a Linear Programming Problem A linear programming problem is in standard form if it seeks to maximize the objective function z c x c x... c n x n subject to the constraints a x a x... a n x n b a x a x... a n x n b. a m x a m x... a mn x n b m where x i and b i. After adding slack variables, the corresponding system of constraint equations is a x a x... a n x n s b a x a x... an x n s b. a m x a m x... a mn x n where s i. s m b m
2 SECTION 9. THE SIMPLEX METHOD: MAXIMIZATION 495 REMARK: Note that for a linear programming problem in standard form, the objective function is to be maximized, not minimized. (Minimization problems will be discussed in Sections 9.4 and 9.5.) A basic solution of a linear programming problem in standard form is a solution x, x,..., x n, s, s,..., s m of the constraint equations in which at most m variables are nonzero the variables that are nonzero are called basic variables. A basic solution for which all variables are nonnegative is called a basic feasible solution. The Simplex Tableau The simplex method is carried out by performing elementary row operations on a matrix that we call the simplex tableau. This tableau consists of the augmented matrix corresponding to the constraint equations together with the coefficients of the objective function written in the form c x c x... c n x n s s... s m z. In the tableau, it is customary to omit the coefficient of z. For instance, the simplex tableau for the linear programming problem z 4x 6x Objective function x 5x s s s } x 5x s s s Constraints x 5x s s s 9 is as follows. x x s s s b Variables s s 5 9 s 4 6 Current z value For this initial simplex tableau, the basic variables are s, s, and s, and the nonbasic variables (which have a value of zero) are x and x. Hence, from the two columns that are farthest to the right, we see that the current solution is x, x, s, s, and s 9. This solution is a basic feasible solution and is often written as x, x, s, s, s,,,, 9.
3 496 CHAPTER 9 LINEAR PROGRAMMING The entry in the lower right corner of the simplex tableau is the current value of z. Note that the bottom row entries under x and x are the negatives of the coefficients of x and x in the objective function To perform an optimality check for a solution represented by a simplex tableau, we look at the entries in the bottom row of the tableau. If any of these entries are negative (as above), then the current solution is not optimal. Pivoting z 4x 6x. Once we have set up the initial simplex tableau for a linear programming problem, the simplex method consists of checking for optimality and then, if the current solution is not optimal, improving the current solution. (An improved solution is one that has a larger zvalue than the current solution.) To improve the current solution, we bring a new basic variable into the solution we call this variable the entering variable. This implies that one of the current basic variables must leave, otherwise we would have too many variables for a basic solution we call this variable the departing variable. We choose the entering and departing variables as follows.. The entering variable corresponds to the smallest (the most negative) entry in the bottom row of the tableau.. The departing variable corresponds to the smallest nonnegative ratio of b i a ij, in the column determined by the entering variable.. The entry in the simplex tableau in the entering variable s column and the departing variable s row is called the pivot. Finally, to form the improved solution, we apply GaussJordan elimination to the column that contains the pivot, as illustrated in the following example. (This process is called pivoting.) EXAMPLE Pivoting to Find an Improved Solution Use the simplex method to find an improved solution for the linear programming problem represented by the following tableau. x x s s s b Variables s s 5 9 s 4 6 The objective function for this problem is z 4x 6x.
4 SECTION 9. THE SIMPLEX METHOD: MAXIMIZATION 49 Solution Note that the current solution x, x, s, s, s 9 corresponds to a z value of. To improve this solution, we determine that x is the entering variable, because 6 is the smallest entry in the bottom row. x x s s s b Variables s s 5 9 s 4 6 To see why we choose x as the entering variable, remember that z 4x 6x. Hence, it appears that a unit change in x produces a change of 6 in z, whereas a unit change in x produces a change of only 4 in z. To find the departing variable, we locate the b i s that have corresponding positive elements in the entering variables column and form the following ratios.,, Here the smallest positive ratio is, so we choose s as the departing variable. x x s s s b Variables s Departing s 5 9 s 4 6 Note that the pivot is the entry in the first row and second column. Now, we use Gauss Jordan elimination to obtain the following improved solution. Before Pivoting After Pivoting The new tableau now appears as follows.
5 498 CHAPTER 9 LINEAR PROGRAMMING x x s s s b Variables x 6 s 5 5 s 6 66 Note that has replaced in the basis column and the improved solution x x, x, s, s, s,,, 6, 5 has a zvalue of s z 4x 6x In Example the improved solution is not yet optimal since the bottom row still has a negative entry. Thus, we can apply another iteration of the simplex method to further improve our solution as follows. We choose x as the entering variable. Moreover, the smallest nonnegative ratio of, 6 8, and 5 5 is 5, so s is the departing variable. GaussJordan elimination produces the following Thus, the new simplex tableau is as follows. x x s s s b Variables 6 x 6 s 5 x In this tableau, there is still a negative entry in the bottom row. Thus, we choose s as the entering variable and s as the departing variable, as shown in the following tableau.
6 SECTION 9. THE SIMPLEX METHOD: MAXIMIZATION 499 x x s s s b Variables 6 x 6 s Departing 5 5 x 8 6 By performing one more iteration of the simplex method, we obtain the following tableau. (Try checking this.) x x s s s b Variables x 4 s 5 5 x Maximum zvalue In this tableau, there are no negative elements in the bottom row. We have therefore determined the optimal solution to be with 8 x, x, s, s, s 5,, 4,, z 4x 6x REMARK: Ties may occur in choosing entering and/or departing variables. Should this happen, any choice among the tied variables may be made. Figure 9.8 x 5 (5, 6) 5 (5, ) (, ) 5 (, ) (, ) x Because the linear programming problem in Example involved only two decision variables, we could have used a graphical solution technique, as we did in Example, Section 9.. Notice in Figure 9.8 that each iteration in the simplex method corresponds to moving from a given vertex to an adjacent vertex with an improved zvalue., z The Simplex Method, z 66 5, 6 z 6 5, z We summarize the steps involved in the simplex method as follows.
7 5 CHAPTER 9 LINEAR PROGRAMMING The Simplex Method (Standard Form) To solve a linear programming problem in standard form, use the following steps.. Convert each inequality in the set of constraints to an equation by adding slack variables.. Create the initial simplex tableau.. Locate the most negative entry in the bottom row. The column for this entry is called the entering column. (If ties occur, any of the tied entries can be used to determine the entering column.) 4. Form the ratios of the entries in the bcolumn with their corresponding positive entries in the entering column. The departing row corresponds to the smallest nonnegative ratio b i a ij. (If all entries in the entering column are or negative, then there is no maximum solution. For ties, choose either entry.) The entry in the departing row and the entering column is called the pivot. 5. Use elementary row operations so that the pivot is, and all other entries in the entering column are. This process is called pivoting. 6. If all entries in the bottom row are zero or positive, this is the final tableau. If not, go back to Step.. If you obtain a final tableau, then the linear programming problem has a maximum solution, which is given by the entry in the lowerright corner of the tableau. Note that the basic feasible solution of an initial simplex tableau is x, x,..., x n, s, s,..., s m,,...,, b, b,..., b m. This solution is basic because at most m variables are nonzero (namely the slack variables). It is feasible because each variable is nonnegative. In the next two examples, we illustrate the use of the simplex method to solve a problem involving three decision variables. EXAMPLE The Simplex Method with Three Decision Variables Use the simplex method to find the maximum value of z x x x subject to the constraints x x x x x x x x x 5 Objective function where x, x, and x. Solution Using the basic feasible solution x, x, x, s, s, s,,,,, 5 the initial simplex tableau for this problem is as follows. (Try checking these computations, and note the tie that occurs when choosing the first entering variable.)
8 SECTION 9. THE SIMPLEX METHOD: MAXIMIZATION 5 x x x s s s b Variables s s 5 s Departing x x x s s s b Variables s Departing 5 s x 5 x x x s s s b Variables 5 x 5 s 5 x 5 This implies that the optimal solution is x, x, x, s, s, s 5,, 5,,, and the maximum value of z is 5. 5 Occasionally, the constraints in a linear programming problem will include an equation. In such cases, we still add a slack variable called an artificial variable to form the initial simplex tableau. Technically, this new variable is not a slack variable (because there is no slack to be taken). Once you have determined an optimal solution in such a problem, you should check to see that any equations given in the original constraints are satisfied. Example illustrates such a case. EXAMPLE The Simplex Method with Three Decision Variables Use the simplex method to find the maximum value of z x x x Objective function
9 5 CHAPTER 9 LINEAR PROGRAMMING subject to the constraints 4x x x x x x 6 x x x 4 where x, x, and x. Solution Using the basic feasible solution x, x, x, s, s, s,,,, 6, 4 is an artificial vari the initial simplex tableau for this problem is as follows. (Note that able, rather than a slack variable.) x x x s s s b Variables s 4 s Departing 6 s 4 s x x x s s s b Variables x 5 45 s Departing 65 s x x x s s s b Variables 5 x x s This implies that the optimal solution is x, x, x, s, s, s, 8,,,, 5 45 and the maximum value of z is 45. (This solution satisfies the equation given in the constraints because 4 8.
10 SECTION 9. THE SIMPLEX METHOD: MAXIMIZATION 5 Applications EXAMPLE 4 A Business Application: Maximum Profit A manufacturer produces three types of plastic fixtures. The time required for molding, trimming, and packaging is given in Table 9.. (Times are given in hours per dozen fixtures.) TABLE 9. Process Type A Type B Type C Total time available Molding, Trimming 4,6 Packaging,4 Profit $ $6 $5 How many dozen of each type of fixture should be produced to obtain a maximum profit? Solution Letting x, x, and x represent the number of dozen units of Types A, B, and C, respectively, the objective function is given by Profit P x 6x 5x. Moreover, using the information in the table, we construct the following constraints. x x x, x x x 4,6 x x x,4 (We also assume that x, x, and x. ) Now, applying the simplex method with the basic feasible solution x, x, x, s, s, s,,,,, 4,6,,4 we obtain the following tableaus. x x x s s s b Variables, s Departing 6 5 4,6 s,4 s
11 54 CHAPTER 9 LINEAR PROGRAMMING x x x s s s b Variables 4 6, x 6 s 4 s Departing , x x x s s s b Variables 8 4 5,4 x 4 6 s Departing 4, x ,6 x x x s s s b Variables 5, x x 6 6 x 6 6, From this final simplex tableau, we see that the maximum profit is $,, and this is obtained by the following production levels. Type A: 6 dozen units Type B: 5, dozen units Type C: 8 dozen units REMARK: In Example 4, note that the second simplex tableau contains a tie for the minimum entry in the bottom row. (Both the first and third entries in the bottom row are. ) Although we chose the first column to represent the departing variable, we could have chosen the third column. Try reworking the problem with this choice to see that you obtain the same solution. EXAMPLE 5 A Business Application: Media Selection The advertising alternatives for a company include television, radio, and newspaper advertisements. The costs and estimates for audience coverage are given in Table 9.
12 SECTION 9. THE SIMPLEX METHOD: MAXIMIZATION 55 TABLE 9. Television Newspaper Radio Cost per advertisement $, $ 6 $ Audience per advertisement, 4, 8, The local newspaper limits the number of weekly advertisements from a single company to ten. Moreover, in order to balance the advertising among the three types of media, no more than half of the total number of advertisements should occur on the radio, and at least % should occur on television. The weekly advertising budget is $8,. How many advertisements should be run in each of the three types of media to maximize the total audience? Solution To begin, we let x, x, and x represent the number of advertisements in television, newspaper, and radio, respectively. The objective function (to be maximized) is therefore z,x 4,x 8,x Objective function where x, x, and x. The constraints for this problem are as follows. x 6x x x 6x x x 6x x.5 x x x x 6x x. x x x A more manageable form of this system of constraints is as follows. x 6x x 8 x 6x x } x 6x x 8 Constraints 9x 6x x 8 Thus, the initial simplex tableau is as follows., x x x s s s s 4 b Variables 6 8 s Departing s s 9 s 4 4, 8, 8,
13 56 CHAPTER 9 LINEAR PROGRAMMING Now, to this initial tableau, we apply the simplex method as follows. x x x s s s s 4 b Variables x s Departing s s 4,, 5, 9, x x x s s s s 4 b Variables x x 6 s Departing s 4, 5,,,, x x x s s s s 4 b Variables 9 4 x x 4 4 x s 4 8,, 6,,5, From this tableau, we see that the maximum weekly audience for an advertising budget of $8, is z,5, 4 Maximum weekly audience and this occurs when x 4, x, and x 4. We sum up the results here. Number of Media Advertisements Cost Audience Television 4 $ 8, 4, Newspaper $ 6, 4, Radio 4 $ 4, 5, Total 8 $8,,5,
14 SECTION 9. EXERCISES 5 SECTION 9. EXERCISES In Exercises 4, write the simplex tableau for the given linear programming problem. You do not need to solve the problem. (In each case the objective function is to be maximized.). Objective function:. Objective function: z x x z x x x x 8 x x 4 x x 5 x x x, x x, x. Objective function: 4. Objective function: z x x 4x z 6x 9x x x x x x 6 x x x 8 x x x, x, x x, x In Exercises 5 8, explain why the linear programming problem is not in standard form as given. 5. (Minimize) 6. (Maximize) Objective function: Objective function: z x x z x x x x 4 x x 6 x, x x x x, x. (Maximize) 8. (Maximize) Objective function: Objective function: z x x z x x x x x 5 x x 4 x x x x x 6 x x x x, x x, x, x In Exercises 9, use the simplex method to solve the given linear programming problem. (In each case the objective function is to be maximized.) 9. Objective function:. Objective function: z x x z x x x 4x 8 x x 6 x 4x x x x, x x, x. Objective function:. Objective function: z 5x x 8x z x x x x 4x x 4 x x 8 x x x 4 x x 5 6x x x 4 x, x, x x, x, x 4. Objective function: 4. Objective function: z 4x 5x z x x x x x x 5 x x 4 x x x, x 4 x, x 5. Objective function: 6. Objective function: z x 4x x x 4 z x 8x x 4x 5x 4 x x 6 x 6x 4x 5x 4 x x 8 x 4x 5x x 4 8 x 4x 48 x, x, x, x 4 x, x 4. Objective function: 8. Objective function: z x x x z x x x x x x 4 x x x 59 x x x 5 x x x 5 x x x x x 6x 54 x, x, x x, x, x 5 9. Objective function: z x x x 4 x x x x 4 4 x x x x 4 4 x, x, x, x 4 4. Objective function: z x x x x 4 x x x 4x 4 6 x x x 5x 4 5 x x x 6x 4 x, x, x, x 4
15 58 CHAPTER 9 LINEAR PROGRAMMING. A merchant plans to sell two models of home computers at costs of $5 and $4, respectively. The $5 model yields a profit of $45 and the $4 model yields a profit of $5. The merchant estimates that the total monthly demand will not exceed 5 units. Find the number of units of each model that should be stocked in order to maximize profit. Assume that the merchant does not want to invest more than $, in computer inventory. (See Exercise in Section 9..). A fruit grower has 5 acres of land available to raise two crops, A and B. It takes one day to trim an acre of crop A and two days to trim an acre of crop B, and there are 4 days per year available for trimming. It takes. day to pick an acre of crop A and. day to pick an acre of crop B, and there are days per year available for picking. Find the number of acres of each fruit that should be planted to maximize profit, assuming that the profit is $4 per acre for crop A and $5 per acre for B. (See Exercise in Section 9..). A grower has 5 acres of land for which she plans to raise three crops. It costs $ to produce an acre of carrots and the profit is $6 per acre. It costs $8 to produce an acre of celery and the profit is $ per acre. Finally, it costs $4 to produce an acre of lettuce and the profit is $ per acre. Use the simplex method to find the number of acres of each crop she should plant in order to maximize her profit. Assume that her cost cannot exceed $,. 4. A fruit juice company makes two special drinks by blending apple and pineapple juices. The first drink uses % apple juice and % pineapple, while the second drink uses 6% apple and 4% pineapple. There are liters of apple and 5 liters of pineapple juice available. If the profit for the first drink is $.6 per liter and that for the second drink is $.5, use the simplex method to find the number of liters of each drink that should be produced in order to maximize the profit. 5. A manufacturer produces three models of bicycles. The time (in hours) required for assembling, painting, and packaging each model is as follows. Model A Model B Model C Assembling.5 Painting.5 Packaging.5.5 The total time available for assembling, painting, and packaging is 46 hours, 495 hours and 5 hours, respectively. The profit per unit for each model is $45 (Model A), $5 (Model B), and $55 (Model C). How many of each type should be produced to obtain a maximum profit? 6. Suppose in Exercise 5 the total time available for assembling, painting, and packaging is 4 hours, 5 hours, and 5 hours, respectively, and that the profit per unit is $48 (Model A), $5 (Model B), and $5 (Model C). How many of each type should be produced to obtain a maximum profit?. A company has budgeted a maximum of $6, for advertising a certain product nationally. Each minute of television time costs $6, and each onepage newspaper ad costs $5,. Each television ad is expected to be viewed by 5 million viewers, and each newspaper ad is expected to be seen by million readers. The company s market research department advises the company to use at most 9% of the advertising budget on television ads. How should the advertising budget be allocated to maximize the total audience? 8. Rework Exercise assuming that each onepage newspaper ad costs $,. 9. An investor has up to $5, to invest in three types of investments. Type A pays 8% annually and has a risk factor of. Type B pays % annually and has a risk factor of.6. Type C pays 4% annually and has a risk factor of.. To have a wellbalanced portfolio, the investor imposes the following conditions. The average risk factor should be no greater than.5. Moreover, at least onefourth of the total portfolio is to be allocated to Type A investments and at least onefourth of the portfolio is to be allocated to Type B investments. How much should be allocated to each type of investment to obtain a maximum return?. An investor has up to $45, to invest in three types of investments. Type A pays 6% annually and has a risk factor of. Type B pays % annually and has a risk factor of.6. Type C pays % annually and has a risk factor of.8. To have a wellbalanced portfolio, the investor imposes the following conditions. The average risk factor should be no greater than.5. Moreover, at least onehalf of the total portfolio is to be allocated to Type A investments and at least onefourth of the portfolio is to be allocated to Type B investments. How much should be allocated to each type of investment to obtain a maximum return?. An accounting firm has 9 hours of staff time and hours of reviewing time available each week. The firm charges $ for an audit and $ for a tax return. Each audit requires hours of staff time and hours of review time, and each tax return requires.5 hours of staff time and.5 hours of review time. What number of audits and tax returns will bring in a maximum revenue?
16 SECTION 9.4 THE SIMPLEX METHOD: MINIMIZATION 59. The accounting firm in Exercise raises its charge for an audit to $5. What number of audits and tax returns will bring in a maximum revenue? In the simplex method, it may happen that in selecting the departing variable all the calculated ratios are negative. This indicates an unbounded solution. Demonstrate this in Exercises and 4.. (Maximize) 4. (Maximize) Objective function: Objective function: z x x z x x x x x x x x 4 x x 5 x, x x, x 5 If the simplex method terminates and one or more variables not in the final basis have bottomrow entries of zero, bringing these variables into the basis will determine other optimal solutions. Demonstrate this in Exercises 5 and (Maximize) 6. (Maximize) Objective function: Objective function: z.5x z x x x x 5x 5 x x 5x x x x 5 x, x x, x C. Use a computer to maximize the objective function z x x 6x 4x 4 subject to the constraints.x.x.8x.5x 4 65.x.x.8x.x x.x.x.4x 4 8 where x, x, x, x 4. C 8. Use a computer to maximize the objective function z.x x x x 4 subject to the same set of constraints given in Exercise. 9.4 THE SIMPLEX METHOD: MINIMIZATION In Section 9., we applied the simplex method only to linear programming problems in standard form where the objective function was to be maximized. In this section, we extend this procedure to linear programming problems in which the objective function is to be minimized. A minimization problem is in standard form if the objective function... w c x c x c n x n is to be minimized, subject to the constraints a x a x... a n x n b a x a x... a n x n b a m x a m x... a mn x n b m. where x i and b i. The basic procedure used to solve such a problem is to convert it to a maximization problem in standard form, and then apply the simplex method as discussed in Section 9.. In Example 5 in Section 9., we used geometric methods to solve the following minimization problem.
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